I entered the territory of Los Angeles, USA, but I was given three no no aos cos eos, and I can only stay for one month [not yet the end].

Those three just can't apply for identity extension, identity change and identity adjustment (green card).

Applying for asylum doesn't even care whether you enter the country legally, but only requires you to apply within one year after you enter the country. Of course, I suggest you consult a reliable immigration lawyer first to see if you really meet the conditions for applying for asylum. Otherwise, it is a waste of time.